RBF-FD schemes for option valuation under models with price-dependent and stochastic volatility

Radial basis functions based finite difference schemes for the solution of partial differential equations have the advantage that an optimal choice of the shape parameter can yield better accuracies than standard finite difference discretisations based on the same number of nodal points. Such schemes known as local radial basis functions methods are considered for the pricing of options under the constant elasticity of variance and the Heston stochastic volatility model. A general methodology for approximating first and second order derivative terms in the finance pdes is presented and the resulting schemes are applied for option valuation. For one-dimensional problems, we derive a compact-RBF scheme which achieves a higher order accuracy when combined with a local mesh refinement strategy. Numerical results and comparisons made for European, American and barrier options illustrate the good performances of the localized radial basis functions methods.